Copyright © University of Cambridge. All rights reserved.

'Square Pair' printed from https://nrich.maths.org/

Show menu

 

Why do this problem?

This problem builds students' understanding of matrix transformations in two dimensions and encourages exploration which will increase confidence at working with vectors and matrices. Insight gained from geometrical approaches leads to a better understanding of matrix algebra.

 

Possible approach

Students could explore this problem by choosing a variety of different matrices and plotting the points which $(0,0), (0,1), (1,0)$ and $(1,1)$ are transformed to. Once they have built up a picture of how the square $S$ is transformed, they could be challenged to find matrices which would transform $S$ into a particular type of quadrilateral, and then to justify why some types of quadrilateral cannot be made by transforming $S$. Students may start by justifying these conjectures using their geometrical insights, but they should be encouraged to support this using appropriate matrix algebra.

 

The second part of the problem asks students to investigate transformations of a second square; this could be done in the same way, first by trying some numerical examples by choosing suitable matrices, then generalising from what they find and supporting their generalisations algebraically.

 

Students might like to use this Matrix Transformation tool to help them investigate the problem.    In this tool the four corners of a quadrilateral are given as a $2 \times 4$ matrix, where the coordinates appear as the columns of the matrix, in clockwise (or anticlockwise) order.

 

Key questions

What can you say about the image of the points on a line after transformation by a matrix?

What can you say about the image of a pair of parallel lines after transformation by a matrix?

 

Possible extension

Transformations for 10 offers a variety of challenging questions about the effects of matrices in two and three dimensions, with an emphasis on thinking geometrically.

 
There are more matrix problems in this feature. 

 

Possible support

Begin with lots of examples of transforming the points $(0,0), (0,1), (1,0), (1,1)$ by multiplying by different matrices. Plot the resulting four points each time, and share ideas about what is common to all the images.

 
The problem Flipping Twisty Matrices starts by considering some specific matrices and seeing what affect these have.